Odds and Probability and How the Two Compare

What is probability, what are odds, and how are the two related?

Probability is the chance that something will happen. It can be expressed as a fraction, a decimal, or a percent. If something is expected to succeed one time out of two, its probability is $1/2=0.5=50%$ .

Odds compares the chance that a thing will happen with the chance that it will not. Odds is usually expressed as a ratio — e.g., $x:y$ . That means the same thing as $\frac{x}{y}$ ; the difference is just formatting. If something is expected to succeed one time out of two, that means one success for one failure, and its odds are 1:1.

In a statistics course, you may hear more about probability than you do about odds. In real life, you probably hear both: the probability of rain today and the odds that your horse will win. Statisticians use both probability and odds.

How do odds and probability compare?

If something is absolutely not going to happen, its odds and its probability are both zero.
You can’t get less likely than absolutely not going to happen: Neither odds nor probability can be negative.
If something is certain to happen:
- Its probability is one; that is, 100%.
- The chance that it will not happen is zero, so odds, the chance that it will happen v. the chance that it won’t, is infinite.

More specifically:

Say $P$ is the chance that something will happen and $N$ is the chance that it will not. Odds, the ratio of the chance that a thing will happen to the chance that it will not, is $P/N$ . Call odds C. Then $O=P/N$ .